A battery is connected between two points `A and B` on the circumference of a uniform conducting ring of radius `r` and resistance `R` . One of the arcs `AB` of the ring subtends an angle `theta` at the centre . The value of the magnetic induction at the centre due to the current in the ring is

Options

A. proportional to `2(180^@- theta)`
B. inversely proportional to r
C. zero, only if `theta=180^@`
D. zero for all values `theta`

Answer

D

Explanation

(d) Magnetic field at the centre due to current in arc ACB is
`B_1=(mu_0)/(4pi) (2piI_1)/r theta/(2pi)` (directed upwards)
magnetic field at the centre due to current in arc ADB is
`B_2=(mu_0)/(4pi) (2piI_2)/r (2pi-theta)/(2pi)` (directed downwards)

Therefore, net magnetic field at the centre
`B=(mu_0)/(4pi) (2piI_1)/r theta/(2pi)-(mu_0)/(4pi) (2piI_2)/r (2pi-theta)/(2pi)`
Also, `I_1=E/(R_1)=E/(rhol_1//A)=(EA)/(rhor theta)`
And `I_2=E/(R_2)=E/(rhol_2//A)=(EA)/(rhor(2pi- theta))`
`:. B=(mu_0)/(4pi)[(EA)/(rhor theta)xxtheta/r-(EA)/(rhor(2pi-theta))xx((2pi-theta))/r]=0`
(d) is the correct option.